Question

Let \(X\) be \(b(2, p)\) and let \(Y\) be \(b(4, p)\). If \(P(X \geq 1)=\frac{5}{9}\), find \(P(Y \geq 1)\).

Short answer

The probability \(P(Y \geq 1)\) is \(\frac{15}{16}\).

Step-by-step solution

Calculate value of p

Given \(P(X ≥ 1) = \frac{5}{9}\), we can calculate \(P(X = 0)\) as it is easier and then subtract it from 1. Because X is binomially distributed, \(P(X = 0) = (1 - p)^2\). Hence, 1 - \(P(X = 0)\) = \(P(X ≥ 1)\). Solve this equation to find \(p\).

Apply Binomial Theorem

Applying binomial theorem, we derive \(p = 1 - \sqrt{1 - \frac{5}{9}} = \frac{2}{3}\)

Calculate \(P(Y ≥ 1)\)

Now with the same approach, \(P(Y ≥ 1)\) = 1 - \(P(Y = 0)\). As we already calculated, \(p = \frac{2}{3}\). Summarising, \(P(Y = 0)\) = \((1 - p)^4 = (1 - \frac{2}{3})^4 = \frac{1}{16}\). Hence, \(P(Y ≥ 1)\) = 1 - \((1 - p)^4 = 1 - \frac{1}{16} = \frac{15}{16}\).